Question: Solve the equation. $\dfrac{dy}{dx}=\dfrac{x^2}{9}+\dfrac{7}{9}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\dfrac{2x}{9}+C$ (Choice B) B $y=\dfrac{x}{18}+C$ (Choice C) C $y=\dfrac{x^3}{3}+\dfrac{7x}{9}+C$ (Choice D) D $y=\dfrac{x^3}{27}+\dfrac{7x}{9}+C$
Solution: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{x^2}{9}+\dfrac{7}{9} \\\\ \dfrac{dy}{dx}&=\dfrac{1}{9}(x^2+7) \\\\ 9\,dy&=(x^2+7)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} 9\,dy&=(x^2+7)\,dx \\\\ \int 9\,dy&=\int (x^2+7)\,dx \\\\ 9y&=\dfrac{x^3}{3}+7x+C_1 \\\\ y&=\dfrac{x^3}{27}+\dfrac{7x}{9}+C \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\dfrac{x^3}{27}+\dfrac{7x}{9}+C$